Gray-Scott 2D

anypinn create my-project --template gray-scott-2d

Coupled two-field reaction-diffusion PDE. Recovers diffusion rates and reaction parameters from pattern snapshots.

Background

The Gray-Scott model describes two reacting and diffusing chemical species in a continuously stirred tank reactor: a substrate \(u\) is fed at rate \(F\) and consumed by an autocatalytic reaction \(u + 2v \to 3v\), while the product \(v\) decays at rate \(F + k\). Depending on the feed rate \(F\) and kill rate \(k\), the system produces a rich variety of Turing-type patterns including spots, stripes, labyrinthine structures, and self-replicating pulses. It is widely studied as a prototype for self-organizing pattern formation in chemistry and developmental biology.

Governing Equations

\[ \begin{cases} \dfrac{\partial u}{\partial t} = D_u\,\nabla^2 u - u\,v^2 + F\,(1 - u) \\[8pt] \dfrac{\partial v}{\partial t} = D_v\,\nabla^2 v + u\,v^2 - (F + k)\,v \end{cases} \]

where:

• \(u(\mathbf{x}, t)\): substrate concentration
• \(v(\mathbf{x}, t)\): product concentration
• \(D_u, D_v\): diffusion coefficients (to recover)
• \(F\): feed rate (to recover)
• \(k\): kill rate (to recover)
• \(\nabla^2 = \dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2}\): 2D Laplacian

Default Configuration

The generated template uses the following values.

Parameters to recover:

Symbol	Code constant	True value
\(D_u\)	TRUE_DU	\(5 \times 10^{-3}\)
\(D_v\)	TRUE_DV	\(2.5 \times 10^{-3}\)
\(F\)	TRUE_F	\(0.04\)
\(k\)	TRUE_K	\(0.06\)

Initial conditions: \(u = 1,\; v = 0\) everywhere, except a central square \([0.4, 0.6]^2\) where \(u = 0.5,\; v = 0.25\) (seeding the pattern).

Domain: \((x, y) \in [0, 1]^2, \quad t \in [0, 200]\)

Boundary conditions: Neumann (zero-flux) on all edges.

Features Demonstrated

• 2D PDE with coupled fields
• PDEResidualConstraint with field-subset scoping
• Multi-parameter recovery (diffusion rates and reaction parameters)

Results